# 754 and Level 1

• 754 is a composite number.
• Prime factorization: 754 = 2 x 13 x 29
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 754 has exactly 8 factors.
• Factors of 754: 1, 2, 13, 26, 29, 58, 377, 754
• Factor pairs: 754 = 1 x 754, 2 x 377, 13 x 58, or 26 x 29
• 754 has no square factors that allow its square root to be simplified. √754 ≈ 27.459060. Here’s today’s puzzle: Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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I have much more to tell you about the number 754:

754 is the sum of consecutive numbers three different ways:

• 187 + 188 + 189 + 190 = 754; that’s 4 consecutive numbers.
• 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 = 754; that’s 13 consecutive numbers.
• 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 = 754; that’s 29 consecutive numbers.

Because all of the odd prime factors of 754 can be expressed as the sum of two squares, 754 can also be expressed as the sum of two squares:

• 27² + 5² = 754
• 23² + 15² = 754

Also because two of its prime factors are hypotenuses of primitive Pythagorean triples, I knew that 754 is the hypotenuse of FOUR Pythagorean triples:

• 270² + 704² = 754²; the triple 270-704-754 was calculated from 2(27)(5), 27² – 5², 27² + 5².
• 290² + 696² = 754²
• 304² + 690² = 754²; the triple 304-690-754 was calculated from 23² – 15², 2(23)(15), 23² + 15².
• 520² + 546² = 754²

754 can also be written as the sum of three squares four different ways:

• 27² + 4² + 3² = 754
• 24² + 13² + 3² = 754
• 23² + 12² + 9² = 754
• 21² + 13² + 12² = 754

754 is a palindrome in three bases:

• 626 BASE 11; note that 6(121) + 2(11) + 6(1) = 754.
• 2F2 BASE 16 (F is 15 base 10); note that 2(256) + 15(16) + 2(1) = 754.
• QQ BASE 28(Q is 26 base 10); note that 26(28) + 26(1) = 754.

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